CMPINF 2100 Fall 2021 - Week 08

PCA with the Sonar data set

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

import seaborn as sns

Sonar data set

sonar_url = 'https://archive.ics.uci.edu/ml/machine-learning-databases/undocumented/connectionist-bench/sonar/sonar.all-data'

sonar_df = pd.read_csv( sonar_url, header=None, prefix='X' )

print( sonar_df.shape )

(208, 61)

sonar_df.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 208 entries, 0 to 207
Data columns (total 61 columns):
 #   Column  Non-Null Count  Dtype  
---  ------  --------------  -----  
 0   X0      208 non-null    float64
 1   X1      208 non-null    float64
 2   X2      208 non-null    float64
 3   X3      208 non-null    float64
 4   X4      208 non-null    float64
 5   X5      208 non-null    float64
 6   X6      208 non-null    float64
 7   X7      208 non-null    float64
 8   X8      208 non-null    float64
 9   X9      208 non-null    float64
 10  X10     208 non-null    float64
 11  X11     208 non-null    float64
 12  X12     208 non-null    float64
 13  X13     208 non-null    float64
 14  X14     208 non-null    float64
 15  X15     208 non-null    float64
 16  X16     208 non-null    float64
 17  X17     208 non-null    float64
 18  X18     208 non-null    float64
 19  X19     208 non-null    float64
 20  X20     208 non-null    float64
 21  X21     208 non-null    float64
 22  X22     208 non-null    float64
 23  X23     208 non-null    float64
 24  X24     208 non-null    float64
 25  X25     208 non-null    float64
 26  X26     208 non-null    float64
 27  X27     208 non-null    float64
 28  X28     208 non-null    float64
 29  X29     208 non-null    float64
 30  X30     208 non-null    float64
 31  X31     208 non-null    float64
 32  X32     208 non-null    float64
 33  X33     208 non-null    float64
 34  X34     208 non-null    float64
 35  X35     208 non-null    float64
 36  X36     208 non-null    float64
 37  X37     208 non-null    float64
 38  X38     208 non-null    float64
 39  X39     208 non-null    float64
 40  X40     208 non-null    float64
 41  X41     208 non-null    float64
 42  X42     208 non-null    float64
 43  X43     208 non-null    float64
 44  X44     208 non-null    float64
 45  X45     208 non-null    float64
 46  X46     208 non-null    float64
 47  X47     208 non-null    float64
 48  X48     208 non-null    float64
 49  X49     208 non-null    float64
 50  X50     208 non-null    float64
 51  X51     208 non-null    float64
 52  X52     208 non-null    float64
 53  X53     208 non-null    float64
 54  X54     208 non-null    float64
 55  X55     208 non-null    float64
 56  X56     208 non-null    float64
 57  X57     208 non-null    float64
 58  X58     208 non-null    float64
 59  X59     208 non-null    float64
 60  X60     208 non-null    object 
dtypes: float64(60), object(1)
memory usage: 99.2+ KB

sonar_df

	X0	X1	X2	X3	X4	X5	X6	X7	X8	X9	...	X51	X52	X53	X54	X55	X56	X57	X58	X59	X60
0	0.0200	0.0371	0.0428	0.0207	0.0954	0.0986	0.1539	0.1601	0.3109	0.2111	...	0.0027	0.0065	0.0159	0.0072	0.0167	0.0180	0.0084	0.0090	0.0032	R
1	0.0453	0.0523	0.0843	0.0689	0.1183	0.2583	0.2156	0.3481	0.3337	0.2872	...	0.0084	0.0089	0.0048	0.0094	0.0191	0.0140	0.0049	0.0052	0.0044	R
2	0.0262	0.0582	0.1099	0.1083	0.0974	0.2280	0.2431	0.3771	0.5598	0.6194	...	0.0232	0.0166	0.0095	0.0180	0.0244	0.0316	0.0164	0.0095	0.0078	R
3	0.0100	0.0171	0.0623	0.0205	0.0205	0.0368	0.1098	0.1276	0.0598	0.1264	...	0.0121	0.0036	0.0150	0.0085	0.0073	0.0050	0.0044	0.0040	0.0117	R
4	0.0762	0.0666	0.0481	0.0394	0.0590	0.0649	0.1209	0.2467	0.3564	0.4459	...	0.0031	0.0054	0.0105	0.0110	0.0015	0.0072	0.0048	0.0107	0.0094	R
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
203	0.0187	0.0346	0.0168	0.0177	0.0393	0.1630	0.2028	0.1694	0.2328	0.2684	...	0.0116	0.0098	0.0199	0.0033	0.0101	0.0065	0.0115	0.0193	0.0157	M
204	0.0323	0.0101	0.0298	0.0564	0.0760	0.0958	0.0990	0.1018	0.1030	0.2154	...	0.0061	0.0093	0.0135	0.0063	0.0063	0.0034	0.0032	0.0062	0.0067	M
205	0.0522	0.0437	0.0180	0.0292	0.0351	0.1171	0.1257	0.1178	0.1258	0.2529	...	0.0160	0.0029	0.0051	0.0062	0.0089	0.0140	0.0138	0.0077	0.0031	M
206	0.0303	0.0353	0.0490	0.0608	0.0167	0.1354	0.1465	0.1123	0.1945	0.2354	...	0.0086	0.0046	0.0126	0.0036	0.0035	0.0034	0.0079	0.0036	0.0048	M
207	0.0260	0.0363	0.0136	0.0272	0.0214	0.0338	0.0655	0.1400	0.1843	0.2354	...	0.0146	0.0129	0.0047	0.0039	0.0061	0.0040	0.0036	0.0061	0.0115	M

208 rows × 61 columns

sonar_df.rename(columns={'X60': 'response'}, inplace=True)

sonar_df

	X0	X1	X2	X3	X4	X5	X6	X7	X8	X9	...	X51	X52	X53	X54	X55	X56	X57	X58	X59	response
0	0.0200	0.0371	0.0428	0.0207	0.0954	0.0986	0.1539	0.1601	0.3109	0.2111	...	0.0027	0.0065	0.0159	0.0072	0.0167	0.0180	0.0084	0.0090	0.0032	R
1	0.0453	0.0523	0.0843	0.0689	0.1183	0.2583	0.2156	0.3481	0.3337	0.2872	...	0.0084	0.0089	0.0048	0.0094	0.0191	0.0140	0.0049	0.0052	0.0044	R
2	0.0262	0.0582	0.1099	0.1083	0.0974	0.2280	0.2431	0.3771	0.5598	0.6194	...	0.0232	0.0166	0.0095	0.0180	0.0244	0.0316	0.0164	0.0095	0.0078	R
3	0.0100	0.0171	0.0623	0.0205	0.0205	0.0368	0.1098	0.1276	0.0598	0.1264	...	0.0121	0.0036	0.0150	0.0085	0.0073	0.0050	0.0044	0.0040	0.0117	R
4	0.0762	0.0666	0.0481	0.0394	0.0590	0.0649	0.1209	0.2467	0.3564	0.4459	...	0.0031	0.0054	0.0105	0.0110	0.0015	0.0072	0.0048	0.0107	0.0094	R
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
203	0.0187	0.0346	0.0168	0.0177	0.0393	0.1630	0.2028	0.1694	0.2328	0.2684	...	0.0116	0.0098	0.0199	0.0033	0.0101	0.0065	0.0115	0.0193	0.0157	M
204	0.0323	0.0101	0.0298	0.0564	0.0760	0.0958	0.0990	0.1018	0.1030	0.2154	...	0.0061	0.0093	0.0135	0.0063	0.0063	0.0034	0.0032	0.0062	0.0067	M
205	0.0522	0.0437	0.0180	0.0292	0.0351	0.1171	0.1257	0.1178	0.1258	0.2529	...	0.0160	0.0029	0.0051	0.0062	0.0089	0.0140	0.0138	0.0077	0.0031	M
206	0.0303	0.0353	0.0490	0.0608	0.0167	0.1354	0.1465	0.1123	0.1945	0.2354	...	0.0086	0.0046	0.0126	0.0036	0.0035	0.0034	0.0079	0.0036	0.0048	M
207	0.0260	0.0363	0.0136	0.0272	0.0214	0.0338	0.0655	0.1400	0.1843	0.2354	...	0.0146	0.0129	0.0047	0.0039	0.0061	0.0040	0.0036	0.0061	0.0115	M

208 rows × 61 columns

sonar_df.response = sonar_df.response.astype('category')

sonar_df.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 208 entries, 0 to 207
Data columns (total 61 columns):
 #   Column    Non-Null Count  Dtype   
---  ------    --------------  -----   
 0   X0        208 non-null    float64 
 1   X1        208 non-null    float64 
 2   X2        208 non-null    float64 
 3   X3        208 non-null    float64 
 4   X4        208 non-null    float64 
 5   X5        208 non-null    float64 
 6   X6        208 non-null    float64 
 7   X7        208 non-null    float64 
 8   X8        208 non-null    float64 
 9   X9        208 non-null    float64 
 10  X10       208 non-null    float64 
 11  X11       208 non-null    float64 
 12  X12       208 non-null    float64 
 13  X13       208 non-null    float64 
 14  X14       208 non-null    float64 
 15  X15       208 non-null    float64 
 16  X16       208 non-null    float64 
 17  X17       208 non-null    float64 
 18  X18       208 non-null    float64 
 19  X19       208 non-null    float64 
 20  X20       208 non-null    float64 
 21  X21       208 non-null    float64 
 22  X22       208 non-null    float64 
 23  X23       208 non-null    float64 
 24  X24       208 non-null    float64 
 25  X25       208 non-null    float64 
 26  X26       208 non-null    float64 
 27  X27       208 non-null    float64 
 28  X28       208 non-null    float64 
 29  X29       208 non-null    float64 
 30  X30       208 non-null    float64 
 31  X31       208 non-null    float64 
 32  X32       208 non-null    float64 
 33  X33       208 non-null    float64 
 34  X34       208 non-null    float64 
 35  X35       208 non-null    float64 
 36  X36       208 non-null    float64 
 37  X37       208 non-null    float64 
 38  X38       208 non-null    float64 
 39  X39       208 non-null    float64 
 40  X40       208 non-null    float64 
 41  X41       208 non-null    float64 
 42  X42       208 non-null    float64 
 43  X43       208 non-null    float64 
 44  X44       208 non-null    float64 
 45  X45       208 non-null    float64 
 46  X46       208 non-null    float64 
 47  X47       208 non-null    float64 
 48  X48       208 non-null    float64 
 49  X49       208 non-null    float64 
 50  X50       208 non-null    float64 
 51  X51       208 non-null    float64 
 52  X52       208 non-null    float64 
 53  X53       208 non-null    float64 
 54  X54       208 non-null    float64 
 55  X55       208 non-null    float64 
 56  X56       208 non-null    float64 
 57  X57       208 non-null    float64 
 58  X58       208 non-null    float64 
 59  X59       208 non-null    float64 
 60  response  208 non-null    category
dtypes: category(1), float64(60)
memory usage: 97.9 KB

sonar_df.response.cat.categories

Index(['M', 'R'], dtype='object')

sonar_df.response.value_counts()

M    111
R     97
Name: response, dtype: int64

sns.set_style('whitegrid')

sns.catplot(data = sonar_df, x='response', kind='count')

plt.show()

Explore the continuous features.

sns.displot(data = sonar_df, x='X0', kind='hist')

plt.show()

sns.displot(data = sonar_df, x='X41', kind='hist')

plt.show()

Let's use separate subplots for every column of the numeric features.

fig, ax = plt.subplots(figsize=(18, 12))

sonar_df.hist(ax=ax)

plt.show()

C:\Users\jyurk\AppData\Local\Temp/ipykernel_20508/3777916670.py:3: UserWarning: To output multiple subplots, the figure containing the passed axes is being cleared
  sonar_df.hist(ax=ax)

Use the default wide-format plotting methods in Seaborn.

sns.displot(data = sonar_df, kind='kde', aspect=2)

plt.show()

Make a Seaborn figure with sepearate histograms per facet for each variable.

Reshape from wide to long-format to be able to make use of facets.

sonar_features = sonar_df.select_dtypes('number').copy()

feature_names = sonar_features.columns

lf = sonar_df.melt(id_vars=['response'], value_vars=feature_names, ignore_index=True)

lf.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 12480 entries, 0 to 12479
Data columns (total 3 columns):
 #   Column    Non-Null Count  Dtype   
---  ------    --------------  -----   
 0   response  12480 non-null  category
 1   variable  12480 non-null  object  
 2   value     12480 non-null  float64 
dtypes: category(1), float64(1), object(1)
memory usage: 207.4+ KB

lf

	response	variable	value
0	R	X0	0.0200
1	R	X0	0.0453
2	R	X0	0.0262
3	R	X0	0.0100
4	R	X0	0.0762
...	...	...	...
12475	M	X59	0.0157
12476	M	X59	0.0067
12477	M	X59	0.0031
12478	M	X59	0.0048
12479	M	X59	0.0115

12480 rows × 3 columns

lf.variable.value_counts()

X0     208
X1     208
X32    208
X33    208
X34    208
X35    208
X36    208
X37    208
X38    208
X39    208
X40    208
X41    208
X42    208
X43    208
X44    208
X45    208
X46    208
X47    208
X48    208
X49    208
X50    208
X51    208
X52    208
X53    208
X54    208
X55    208
X56    208
X57    208
X58    208
X31    208
X30    208
X29    208
X14    208
X2     208
X3     208
X4     208
X5     208
X6     208
X7     208
X8     208
X9     208
X10    208
X11    208
X12    208
X13    208
X15    208
X28    208
X16    208
X17    208
X18    208
X19    208
X20    208
X21    208
X22    208
X23    208
X24    208
X25    208
X26    208
X27    208
X59    208
Name: variable, dtype: int64

len( lf.variable.value_counts() )

60

sns.displot(data = lf, x='value', col='variable', col_wrap=8)

plt.show()

To get greater control over the facets we need to use sns.FacetGrid().

g = sns.FacetGrid(data = lf, col='variable', col_wrap=8, sharex=False, sharey=False)

g.map_dataframe(sns.histplot, x='value')

plt.show()

g = sns.FacetGrid(data = lf, col='variable', col_wrap=8, sharex=False, sharey=False)

g.map_dataframe(sns.kdeplot, x='value', hue='response', common_norm=False)

plt.show()

Boxplots makes it easier to directly compare the summary statistics. Examine the summary statistics per continuous variable regardless of the categorical response.

sns.catplot(data = sonar_df, kind='box', aspect=3.5)

plt.show()

Make a boxplot comparing the variable distribution summary statistics grouped by response.

sns.catplot(data = lf, x='variable', y='value', hue='response', kind='box', aspect=3.5)

plt.show()

Focusing just on the average value of the variable grouped by response. The point plot will show the average and the confidence interval on the average.

sns.catplot(data = lf, x='variable', y='value', hue='response', kind='point', ci=95, join=False, aspect=3.5)

plt.show()

Variable relationships

The correlation plot to examine which variables are related to each other.

fig, ax = plt.subplots(figsize=(12, 8))

sns.heatmap(data = sonar_df.corr(),
            vmin=-1, vmax=1, center=0,
            cmap='coolwarm', 
            ax=ax)

plt.show()

Does the correlation structure change across groups? Look at the correlation plot between the continuous features for each value of response.

response_values = sonar_df.response.cat.categories.tolist()

response_values

['M', 'R']

What does enumerate() do?

for index_value, response_name in enumerate(response_values):
    print( index_value )
    print( response_name )
    print( '-- -- --' )

0
M
-- -- --
1
R
-- -- --

print( response_values[0] )

print( response_values[1] )

M
R

Apply the .corr() method after grouping by response.

sonar_df.groupby(['response']).corr()

		X0	X1	X2	X3	X4	X5	X6	X7	X8	X9	...	X50	X51	X52	X53	X54	X55	X56	X57	X58	X59
response
M	X0	1.000000	0.783947	0.607217	0.538198	0.374794	0.265313	0.248499	0.317525	0.269724	0.232297	...	0.247820	0.383970	0.335548	0.369332	0.378374	0.216367	0.368172	0.365448	0.435246	0.411431
	X1	0.783947	1.000000	0.798240	0.621277	0.404582	0.285976	0.189745	0.260391	0.215417	0.127245	...	0.316439	0.493794	0.379255	0.394056	0.390750	0.218963	0.345689	0.319436	0.389355	0.393674
	X2	0.607217	0.798240	1.000000	0.808222	0.573819	0.301853	0.079446	0.073000	0.056679	-0.001124	...	0.227444	0.440850	0.385773	0.401487	0.309270	0.260026	0.413314	0.291909	0.469410	0.403399
	X3	0.538198	0.621277	0.808222	1.000000	0.756090	0.263246	0.143279	0.088565	0.055072	0.009118	...	0.093955	0.357852	0.438598	0.344309	0.284669	0.292302	0.372026	0.336071	0.491601	0.477271
	X4	0.374794	0.404582	0.573819	0.756090	1.000000	0.472487	0.173684	-0.015053	-0.072556	-0.089119	...	0.024579	0.232731	0.402319	0.219546	0.198967	0.194241	0.289188	0.239418	0.315575	0.287459
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
R	X55	0.139376	0.296085	0.294758	0.175723	0.141973	0.276164	0.241161	0.279460	0.101503	0.078120	...	0.098002	0.028793	0.314731	0.211119	0.245662	1.000000	0.425075	0.287584	0.177656	0.284521
	X56	0.271153	0.197175	0.360617	0.341398	0.131243	0.150329	0.189387	0.362243	0.352957	0.335899	...	0.023743	0.028285	0.093473	0.172256	0.131229	0.425075	1.000000	0.402218	0.208971	0.116897
	X57	0.245461	0.339230	0.357181	0.247430	0.135547	0.080440	0.088492	0.187300	0.105230	0.146996	...	0.220567	0.157240	0.240337	0.202872	0.246434	0.287584	0.402218	1.000000	0.449847	0.340206
	X58	0.068473	0.200830	0.275996	0.175349	0.183776	0.134369	0.113277	0.191932	0.065525	0.123355	...	0.248289	0.069700	0.243011	0.226194	0.225749	0.177656	0.208971	0.449847	1.000000	0.515609
	X59	0.081974	0.217208	0.257229	0.115923	0.135659	0.143853	0.158014	0.214041	-0.039056	0.105013	...	0.347218	0.118140	0.262433	0.219505	0.228338	0.284521	0.116897	0.340206	0.515609	1.000000

120 rows × 60 columns

fig, axs = plt.subplots(1, len(response_values), figsize=(24, 10))

for ix, aname in enumerate(response_values):
    sns.heatmap(data = sonar_df.groupby(['response']).corr().loc[aname],
                vmin=-1, vmax=1, center=0,
                cmap='coolwarm', cbar=False,
                ax=axs[ix])
    
    axs[ix].set_title('response: '+aname)

plt.show()

PCA

Principal Components Analysis (PCA) exploits correlation between variables. New variables are created accounting for all of the original variables.

Before running PCA, we should standardize the variables.

from sklearn.preprocessing import StandardScaler

from sklearn.decomposition import PCA

Standardize the continuous variables.

Xsonar = StandardScaler().fit_transform( sonar_features )

print( type(Xsonar) )

print( Xsonar.shape )

<class 'numpy.ndarray'>
(208, 60)

Check that all variables have roughly the same range.

sns.catplot(data = pd.DataFrame(Xsonar, columns=sonar_features.columns), kind='box', aspect=3.5)

plt.show()

To apply PCA we need to first initialize the object, then fit, and finally transform.

pca_object = PCA()

pca_object.fit( Xsonar )

PCA()

sonar_pcs = pca_object.transform( Xsonar )

print( type(sonar_pcs) )

<class 'numpy.ndarray'>

print( sonar_pcs.shape )

(208, 60)

pc_names = ['pc_'+str(d).zfill(2) for d in range(1, (sonar_pcs.shape[1]+1))]

pc_names

['pc_01',
 'pc_02',
 'pc_03',
 'pc_04',
 'pc_05',
 'pc_06',
 'pc_07',
 'pc_08',
 'pc_09',
 'pc_10',
 'pc_11',
 'pc_12',
 'pc_13',
 'pc_14',
 'pc_15',
 'pc_16',
 'pc_17',
 'pc_18',
 'pc_19',
 'pc_20',
 'pc_21',
 'pc_22',
 'pc_23',
 'pc_24',
 'pc_25',
 'pc_26',
 'pc_27',
 'pc_28',
 'pc_29',
 'pc_30',
 'pc_31',
 'pc_32',
 'pc_33',
 'pc_34',
 'pc_35',
 'pc_36',
 'pc_37',
 'pc_38',
 'pc_39',
 'pc_40',
 'pc_41',
 'pc_42',
 'pc_43',
 'pc_44',
 'pc_45',
 'pc_46',
 'pc_47',
 'pc_48',
 'pc_49',
 'pc_50',
 'pc_51',
 'pc_52',
 'pc_53',
 'pc_54',
 'pc_55',
 'pc_56',
 'pc_57',
 'pc_58',
 'pc_59',
 'pc_60']

sonar_pcs_df = pd.DataFrame( sonar_pcs, columns=pc_names)

sonar_pcs_df

	pc_01	pc_02	pc_03	pc_04	pc_05	pc_06	pc_07	pc_08	pc_09	pc_10	...	pc_51	pc_52	pc_53	pc_54	pc_55	pc_56	pc_57	pc_58	pc_59	pc_60
0	1.921168	-1.370893	-1.666476	0.837913	-1.057324	1.712504	1.785716	-1.581264	0.335418	-1.028065	...	-0.335958	-0.331626	-0.198251	-0.216336	0.097644	-0.136728	-0.004505	-0.095476	0.037942	0.111545
1	-0.480125	7.586388	-1.275734	3.859346	2.121112	-2.186818	-1.742764	1.517061	0.307933	-1.341882	...	-0.148443	-0.072357	-0.236929	0.284475	-0.168813	0.392446	-0.059563	0.284776	0.093821	0.066267
2	3.859228	6.439860	-0.030635	5.454599	1.552060	1.181619	-1.820138	-1.495929	-1.152459	-1.006030	...	-0.217425	0.173613	-0.244956	0.192608	-0.243847	-0.075688	0.234800	-0.167007	-0.064546	0.030407
3	4.597419	-3.104089	-1.785344	-1.115908	-2.785528	-2.072673	2.084530	1.707289	0.452390	-1.117318	...	-0.118437	-0.143758	-0.403720	-0.138827	0.166721	0.169762	0.254800	0.036843	-0.011256	-0.081164
4	-0.533868	1.849847	-0.860097	3.302076	2.808954	-0.783945	0.362657	0.812621	0.184578	-0.023594	...	0.231652	0.407619	0.218464	0.220151	-0.106636	0.119250	-0.153217	0.343264	-0.092512	-0.073373
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
203	-1.207653	-0.968174	3.116148	-0.212490	2.306835	1.151569	0.010206	-1.606494	0.587992	-2.089473	...	-0.092533	0.016097	0.213204	0.013699	0.112502	0.017959	0.122765	0.127351	0.035411	0.045970
204	-2.971439	-2.753492	2.500966	0.478754	2.157206	0.928558	-0.738412	-0.105210	-0.748383	-0.255461	...	0.119349	0.016471	0.161043	0.144495	0.091137	-0.091500	0.063065	0.069433	0.026908	-0.009198
205	-2.293210	-2.755446	2.388467	0.130616	2.759021	1.049497	-0.449578	-1.619202	0.317789	-0.394252	...	0.163217	-0.204229	0.015998	0.096939	0.171822	0.100886	-0.065289	-0.044455	-0.097103	-0.008605
206	-3.114464	-1.850550	2.420486	0.382211	1.658317	0.917819	1.078272	-0.364099	0.536476	0.457437	...	-0.076407	0.142360	-0.095447	-0.276158	0.000244	-0.079732	0.071281	0.082844	0.043121	-0.022540
207	-3.238624	-2.277094	1.706128	-0.225883	1.126790	0.980035	0.672915	-0.625718	-0.553615	0.588364	...	0.153687	0.131752	0.297334	-0.212383	-0.151163	-0.037334	0.039279	-0.047009	-0.102992	0.008450

208 rows × 60 columns

Examine the distributions of the Principal Component scores via boxplots.

sns.catplot(data = sonar_pcs_df, kind='box', aspect=3.5)

plt.show()

The variation is decreasing across the PCs.

fig, ax = plt.subplots(figsize=(12, 8))

sonar_pcs_df.describe().loc['std'].plot(ax=ax)

plt.show()

All of the PCs are uncorrelated.

fig, ax = plt.subplots(figsize=(12, 8))

sns.heatmap(data = sonar_pcs_df.corr(),
            vmin=-1, vmax=1, center=0,
            cmap='coolwarm',
            ax=ax)

plt.show()

We can decide the number of PCs to focus on. There are several hueristics to help us identify the fewest number of PCs to use. One such approach is a visual knee bend via the scree plot.

The scree plot shows the fraction of the variance explained per PC.

fraction_var_explain = pca_object.explained_variance_ratio_

fig, ax = plt.subplots(figsize=(12, 8))

ax.plot( np.arange(len(fraction_var_explain)) + 1, fraction_var_explain, 'o-' )

ax.set_xlabel('PC')
ax.set_ylabel('Variance explained ratio')

plt.show()

From the scree plot it looks 10 PCs are useful.

sns.catplot(data = sonar_pcs_df.iloc[:, :10], kind='box', aspect=3.5)

plt.show()

sns.catplot(data = sonar_pcs_df.iloc[:, :10], kind='violin', aspect=3.5)

plt.show()

Since we have 10 instead of 60 variables we can use the pair plot.

sns.pairplot( data = sonar_pcs_df.iloc[:, :10])

plt.show()

Let's examine the first 10 PCs grouped by response.

sonar_pcs_df['response'] = sonar_df.response

sns.pairplot(data = sonar_pcs_df, 
             vars=sonar_pcs_df.iloc[:, :10].columns,
             hue='response',
             diag_kws={'common_norm': False})

plt.show()